<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
is	O
a	O
binary	O
tree	O
derived	O
from	O
a	O
sequence	O
of	O
numbers	O
.	O
</s>
<s>
When	O
all	O
numbers	O
are	O
distinct	O
,	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
is	O
uniquely	O
defined	O
from	O
the	O
properties	O
that	O
it	O
is	O
heap-ordered	O
and	O
that	O
a	O
symmetric	O
(	O
in-order	O
)	O
traversal	O
of	O
the	O
tree	O
returns	O
the	O
original	O
sequence	O
.	O
</s>
<s>
Introduced	O
by	O
in	O
the	O
context	O
of	O
geometric	O
range	B-Data_Structure
searching	I-Data_Structure
data	B-General_Concept
structures	I-General_Concept
,	O
Cartesian	B-Algorithm
trees	I-Algorithm
have	O
also	O
been	O
used	O
in	O
the	O
definition	O
of	O
the	O
treap	B-Application
and	O
randomized	B-Application
binary	I-Application
search	I-Application
tree	I-Application
data	B-General_Concept
structures	I-General_Concept
for	O
binary	O
search	O
problems	O
,	O
in	O
comparison	B-Algorithm
sort	I-Algorithm
algorithms	O
that	O
perform	O
efficiently	O
on	O
nearly-sorted	O
inputs	O
,	O
and	O
as	O
the	O
basis	O
for	O
pattern	B-Language
matching	I-Language
algorithms	O
.	O
</s>
<s>
A	O
Cartesian	B-Algorithm
tree	I-Algorithm
for	O
a	O
sequence	O
may	O
be	O
constructed	O
in	O
linear	O
time	O
.	O
</s>
<s>
The	O
Cartesian	B-Algorithm
tree	I-Algorithm
for	O
a	O
sequence	O
of	O
distinct	O
numbers	O
is	O
defined	O
by	O
the	O
following	O
properties	O
:	O
</s>
<s>
The	O
Cartesian	B-Algorithm
tree	I-Algorithm
for	O
a	O
sequence	O
is	O
a	O
binary	O
tree	O
with	O
one	O
node	O
for	O
each	O
number	O
in	O
the	O
sequence	O
.	O
</s>
<s>
The	O
tree	O
has	O
the	O
heap	B-Application
property	I-Application
:	O
the	O
parent	O
of	O
any	O
non-root	O
node	O
has	O
a	O
smaller	O
value	O
than	O
the	O
node	O
itself	O
.	O
</s>
<s>
Based	O
on	O
the	O
heap	B-Application
property	I-Application
,	O
the	O
root	O
of	O
the	O
tree	O
must	O
be	O
the	O
smallest	O
number	O
in	O
the	O
sequence	O
.	O
</s>
<s>
From	O
this	O
,	O
the	O
tree	O
itself	O
may	O
also	O
be	O
defined	O
recursively	O
:	O
the	O
root	O
is	O
the	O
minimum	O
value	O
of	O
the	O
sequence	O
,	O
and	O
the	O
left	O
and	O
right	O
subtrees	O
are	O
the	O
Cartesian	B-Algorithm
trees	I-Algorithm
for	O
the	O
subsequences	O
to	O
the	O
left	O
and	O
right	O
of	O
the	O
root	O
value	O
.	O
</s>
<s>
For	O
a	O
sequence	O
of	O
distinct	O
numbers	O
,	O
the	O
three	O
properties	O
above	O
determine	O
a	O
unique	O
Cartesian	B-Algorithm
tree	I-Algorithm
.	O
</s>
<s>
If	O
a	O
sequence	O
of	O
numbers	O
contains	O
repetitions	O
,	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
may	O
be	O
determined	O
for	O
it	O
by	O
following	O
a	O
consistent	O
tie-breaking	O
rule	O
before	O
applying	O
the	O
above	O
construction	O
.	O
</s>
<s>
A	O
Cartesian	B-Algorithm
tree	I-Algorithm
may	O
be	O
constructed	O
in	O
linear	O
time	O
from	O
its	O
input	O
sequence	O
.	O
</s>
<s>
One	O
method	O
is	O
to	O
simply	O
process	O
the	O
sequence	O
values	O
in	O
left-to-right	O
order	O
,	O
maintaining	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
of	O
the	O
nodes	O
processed	O
so	O
far	O
,	O
in	O
a	O
structure	O
that	O
allows	O
both	O
upwards	O
and	O
downwards	O
traversal	O
of	O
the	O
tree	O
.	O
</s>
<s>
The	O
total	O
time	O
for	O
this	O
procedure	O
is	O
linear	O
,	O
because	O
the	O
time	O
spent	O
searching	O
for	O
the	O
parent	O
of	O
each	O
new	O
node	O
can	O
be	O
charged	B-General_Concept
against	O
the	O
number	O
of	O
nodes	O
that	O
are	O
removed	O
from	O
the	O
rightmost	O
path	O
in	O
the	O
tree	O
.	O
</s>
<s>
An	O
alternative	O
linear-time	O
construction	O
algorithm	O
is	O
based	O
on	O
the	O
all	B-Operating_System
nearest	I-Operating_System
smaller	I-Operating_System
values	I-Operating_System
problem	O
.	O
</s>
<s>
The	O
sequence	O
of	O
left	O
neighbors	O
may	O
be	O
found	O
by	O
an	O
algorithm	O
that	O
maintains	O
a	O
stack	B-Application
containing	O
a	O
subsequence	O
of	O
the	O
input	O
.	O
</s>
<s>
For	O
each	O
new	O
sequence	O
value	O
,	O
the	O
stack	B-Application
is	O
popped	O
until	O
it	O
is	O
empty	O
or	O
its	O
top	O
element	O
is	O
smaller	O
than	O
,	O
and	O
then	O
is	O
pushed	O
onto	O
the	O
stack	B-Application
.	O
</s>
<s>
The	O
right	O
neighbors	O
may	O
be	O
found	O
by	O
applying	O
the	O
same	O
stack	B-Application
algorithm	O
to	O
the	O
reverse	O
of	O
the	O
sequence	O
.	O
</s>
<s>
The	O
parent	O
of	O
in	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
is	O
either	O
the	O
left	O
neighbor	O
of	O
or	O
the	O
right	O
neighbor	O
of	O
,	O
whichever	O
exists	O
and	O
has	O
a	O
larger	O
value	O
.	O
</s>
<s>
The	O
left	O
and	O
right	O
neighbors	O
may	O
also	O
be	O
constructed	O
efficiently	O
by	O
parallel	B-Operating_System
algorithms	I-Operating_System
,	O
so	O
this	O
formulation	O
may	O
be	O
used	O
to	O
develop	O
efficient	O
parallel	B-Operating_System
algorithms	I-Operating_System
for	O
Cartesian	B-Algorithm
tree	I-Algorithm
construction	O
.	O
</s>
<s>
Another	O
linear-time	O
algorithm	O
for	O
Cartesian	B-Algorithm
tree	I-Algorithm
construction	O
is	O
based	O
on	O
divide-and-conquer	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
maintain	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
of	O
a	O
dynamic	O
input	O
,	O
subject	O
to	O
insertions	O
of	O
elements	O
and	O
lazy	B-Algorithm
deletion	I-Algorithm
of	O
elements	O
,	O
in	O
logarithmic	O
amortized	B-General_Concept
time	I-General_Concept
per	O
operation	O
.	O
</s>
<s>
Here	O
,	O
lazy	B-Algorithm
deletion	I-Algorithm
means	O
that	O
up	O
to	O
a	O
constant	O
fraction	O
of	O
the	O
elements	O
in	O
the	O
current	O
tree	O
may	O
be	O
marked	O
as	O
deleted	O
,	O
rather	O
than	O
actually	O
removed	O
.	O
</s>
<s>
Cartesian	B-Algorithm
trees	I-Algorithm
may	O
be	O
used	O
as	O
part	O
of	O
an	O
efficient	O
data	B-General_Concept
structure	I-General_Concept
for	O
range	B-Algorithm
minimum	I-Algorithm
queries	I-Algorithm
,	O
a	O
range	B-Data_Structure
searching	I-Data_Structure
problem	O
involving	O
queries	O
that	O
ask	O
for	O
the	O
minimum	O
value	O
in	O
a	O
contiguous	O
subsequence	O
of	O
the	O
original	O
sequence	O
.	O
</s>
<s>
In	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
,	O
this	O
minimum	O
value	O
may	O
be	O
found	O
at	O
the	O
lowest	O
common	O
ancestor	O
of	O
the	O
leftmost	O
and	O
rightmost	O
values	O
in	O
the	O
subsequence	O
.	O
</s>
<s>
Because	O
lowest	O
common	O
ancestors	O
may	O
be	O
found	O
in	O
constant	O
time	O
per	O
query	O
,	O
using	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
takes	O
linear	O
space	O
to	O
store	O
and	O
that	O
may	O
be	O
constructed	O
in	O
linear	O
time	O
,	O
the	O
same	O
bounds	O
hold	O
for	O
the	O
range	O
minimization	O
problem	O
.	O
</s>
<s>
reversed	O
this	O
relationship	O
between	O
the	O
two	O
data	B-General_Concept
structure	I-General_Concept
problems	O
by	O
showing	O
that	O
data	B-General_Concept
structures	I-General_Concept
for	O
range	O
minimization	O
could	O
also	O
be	O
used	O
for	O
finding	O
lowest	O
common	O
ancestors	O
.	O
</s>
<s>
Their	O
data	B-General_Concept
structure	I-General_Concept
associates	O
with	O
each	O
node	O
of	O
the	O
tree	O
its	O
distance	O
from	O
the	O
root	O
,	O
and	O
constructs	O
a	O
sequence	O
of	O
these	O
distances	O
in	O
the	O
order	O
of	O
an	O
Euler	O
tour	O
of	O
the	O
(	O
edge-doubled	O
)	O
tree	O
.	O
</s>
<s>
It	O
then	O
constructs	O
a	O
range	O
minimization	O
data	B-General_Concept
structure	I-General_Concept
for	O
the	O
resulting	O
sequence	O
.	O
</s>
<s>
As	O
they	O
describe	O
,	O
for	O
range	O
minimization	O
in	O
sequences	O
that	O
do	O
not	O
have	O
this	O
form	O
,	O
it	O
is	O
possible	O
to	O
use	O
Cartesian	B-Algorithm
trees	I-Algorithm
to	O
reduce	O
the	O
range	O
minimization	O
problem	O
to	O
lowest	O
common	O
ancestors	O
,	O
and	O
then	O
to	O
use	O
Euler	O
tours	O
to	O
reduce	O
lowest	O
common	O
ancestors	O
to	O
a	O
range	O
minimization	O
problem	O
with	O
this	O
special	O
form	O
.	O
</s>
<s>
The	O
same	O
range	O
minimization	O
problem	O
may	O
also	O
be	O
given	O
an	O
alternative	O
interpretation	O
in	O
terms	O
of	O
two	O
dimensional	O
range	B-Data_Structure
searching	I-Data_Structure
.	O
</s>
<s>
A	O
collection	O
of	O
finitely	O
many	O
points	O
in	O
the	O
Cartesian	O
plane	O
may	O
be	O
used	O
to	O
form	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
,	O
by	O
sorting	B-Algorithm
the	O
points	O
by	O
their	O
-coordinates	O
and	O
using	O
the	O
-coordinates	O
in	O
this	O
order	O
as	O
the	O
sequence	O
of	O
values	O
from	O
which	O
this	O
tree	O
is	O
formed	O
.	O
</s>
<s>
If	O
is	O
the	O
subset	O
of	O
the	O
input	O
points	O
within	O
some	O
vertical	O
slab	O
defined	O
by	O
the	O
inequalities	O
,	O
is	O
the	O
leftmost	O
point	O
in	O
(	O
the	O
one	O
with	O
minimum	O
-coordinate	O
)	O
,	O
and	O
is	O
the	O
rightmost	O
point	O
in	O
(	O
the	O
one	O
with	O
maximum	O
-coordinate	O
)	O
then	O
the	O
lowest	O
common	O
ancestor	O
of	O
and	O
in	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
is	O
the	O
bottommost	O
point	O
in	O
the	O
slab	O
.	O
</s>
<s>
The	O
same	O
construction	O
,	O
of	O
lowest	O
common	O
ancestors	O
in	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
,	O
makes	O
it	O
possible	O
to	O
construct	O
a	O
data	B-General_Concept
structure	I-General_Concept
with	O
linear	O
space	O
that	O
allows	O
the	O
distances	O
between	O
pairs	O
of	O
points	O
in	O
any	O
ultrametric	B-Algorithm
space	I-Algorithm
to	O
be	O
queried	O
in	O
constant	O
time	O
per	O
query	O
.	O
</s>
<s>
The	O
distance	O
within	O
an	O
ultrametric	B-Algorithm
is	O
the	O
same	O
as	O
the	O
minimax	O
path	O
weight	O
in	O
the	O
minimum	O
spanning	O
tree	O
of	O
the	O
metric	O
.	O
</s>
<s>
From	O
the	O
minimum	O
spanning	O
tree	O
,	O
one	O
can	O
construct	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
,	O
the	O
root	O
node	O
of	O
which	O
represents	O
the	O
heaviest	O
edge	O
of	O
the	O
minimum	O
spanning	O
tree	O
.	O
</s>
<s>
Removing	O
this	O
edge	O
partitions	O
the	O
minimum	O
spanning	O
tree	O
into	O
two	O
subtrees	O
,	O
and	O
Cartesian	B-Algorithm
trees	I-Algorithm
recursively	O
constructed	O
for	O
these	O
two	O
subtrees	O
form	O
the	O
children	O
of	O
the	O
root	O
node	O
of	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
.	O
</s>
<s>
The	O
leaves	O
of	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
represent	O
points	O
of	O
the	O
metric	O
space	O
,	O
and	O
the	O
lowest	O
common	O
ancestor	O
of	O
two	O
leaves	O
in	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
is	O
the	O
heaviest	O
edge	O
between	O
those	O
two	O
points	O
in	O
the	O
minimum	O
spanning	O
tree	O
,	O
which	O
has	O
weight	O
equal	O
to	O
the	O
distance	O
between	O
the	O
two	O
points	O
.	O
</s>
<s>
Once	O
the	O
minimum	O
spanning	O
tree	O
has	O
been	O
found	O
and	O
its	O
edge	O
weights	O
sorted	O
,	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
may	O
be	O
constructed	O
in	O
linear	O
time	O
.	O
</s>
<s>
Because	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
is	O
a	O
binary	O
tree	O
,	O
it	O
is	O
natural	O
to	O
use	O
it	O
as	O
a	O
binary	B-Language
search	I-Language
tree	I-Language
for	O
an	O
ordered	O
sequence	O
of	O
values	O
.	O
</s>
<s>
However	O
,	O
defining	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
based	O
on	O
the	O
same	O
values	O
that	O
form	O
the	O
search	O
keys	O
of	O
a	O
binary	B-Language
search	I-Language
tree	I-Language
does	O
not	O
work	O
well	O
:	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
of	O
a	O
sorted	O
sequence	O
is	O
just	O
a	O
path	O
,	O
rooted	O
at	O
its	O
leftmost	O
endpoint	O
,	O
and	O
binary	O
searching	O
in	O
this	O
tree	O
degenerates	O
to	O
sequential	B-Algorithm
search	I-Algorithm
in	O
the	O
path	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
possible	O
to	O
generate	O
more-balanced	O
search	O
trees	O
by	O
generating	O
priority	O
values	O
for	O
each	O
search	O
key	O
that	O
are	O
different	O
than	O
the	O
key	O
itself	O
,	O
sorting	B-Algorithm
the	O
inputs	O
by	O
their	O
key	O
values	O
,	O
and	O
using	O
the	O
corresponding	O
sequence	O
of	O
priorities	O
to	O
generate	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
.	O
</s>
<s>
The	O
data	B-General_Concept
structure	I-General_Concept
resulting	O
from	O
this	O
random	O
choice	O
is	O
called	O
a	O
treap	B-Application
,	O
due	O
to	O
its	O
combination	O
of	O
binary	B-Language
search	I-Language
tree	I-Language
and	O
binary	B-Application
heap	I-Application
features	O
.	O
</s>
<s>
An	O
insertion	O
into	O
a	O
treap	B-Application
may	O
be	O
performed	O
by	O
inserting	O
the	O
new	O
key	O
as	O
a	O
leaf	O
of	O
an	O
existing	O
tree	O
,	O
choosing	O
a	O
priority	O
for	O
it	O
,	O
and	O
then	O
performing	O
tree	B-Data_Structure
rotation	I-Data_Structure
operations	O
along	O
a	O
path	O
from	O
the	O
node	O
to	O
the	O
root	O
of	O
the	O
tree	O
to	O
repair	O
any	O
violations	O
of	O
the	O
heap	B-Application
property	I-Application
caused	O
by	O
this	O
insertion	O
;	O
a	O
deletion	O
may	O
similarly	O
be	O
performed	O
by	O
a	O
constant	O
amount	O
of	O
change	O
to	O
the	O
tree	O
followed	O
by	O
a	O
sequence	O
of	O
rotations	O
along	O
a	O
single	O
path	O
in	O
the	O
tree	O
.	O
</s>
<s>
A	O
variation	O
on	O
this	O
data	B-General_Concept
structure	I-General_Concept
called	O
a	O
zip	O
tree	O
uses	O
the	O
same	O
idea	O
of	O
random	O
priorities	O
,	O
but	O
simplifies	O
the	O
random	O
generation	O
of	O
the	O
priorities	O
,	O
and	O
performs	O
insertions	O
and	O
deletions	O
in	O
a	O
different	O
way	O
,	O
by	O
splitting	O
the	O
sequence	O
and	O
its	O
associated	O
Cartesian	B-Algorithm
tree	I-Algorithm
into	O
two	O
subsequences	O
and	O
two	O
trees	O
and	O
then	O
recombining	O
them	O
.	O
</s>
<s>
If	O
the	O
priorities	O
of	O
each	O
key	O
are	O
chosen	O
randomly	O
and	O
independently	O
once	O
whenever	O
the	O
key	O
is	O
inserted	O
into	O
the	O
tree	O
,	O
the	O
resulting	O
Cartesian	B-Algorithm
tree	I-Algorithm
will	O
have	O
the	O
same	O
properties	O
as	O
a	O
random	O
binary	B-Language
search	I-Language
tree	I-Language
,	O
a	O
tree	O
computed	O
by	O
inserting	O
the	O
keys	O
in	O
a	O
randomly	O
chosen	O
permutation	B-Algorithm
starting	O
from	O
an	O
empty	O
tree	O
,	O
with	O
each	O
insertion	O
leaving	O
the	O
previous	O
tree	O
structure	O
unchanged	O
and	O
inserting	O
the	O
new	O
node	O
as	O
a	O
leaf	O
of	O
the	O
tree	O
.	O
</s>
<s>
Random	O
binary	B-Language
search	I-Language
trees	I-Language
had	O
been	O
studied	O
for	O
much	O
longer	O
,	O
and	O
are	O
known	O
to	O
behave	O
well	O
as	O
search	O
trees	O
(	O
they	O
have	O
logarithmic	O
depth	O
with	O
high	O
probability	O
)	O
;	O
the	O
same	O
good	O
behavior	O
carries	O
over	O
to	O
treaps	B-Application
.	O
</s>
<s>
It	O
is	O
also	O
possible	O
,	O
as	O
suggested	O
by	O
Aragon	O
and	O
Seidel	O
,	O
to	O
reprioritize	O
frequently-accessed	O
nodes	O
,	O
causing	O
them	O
to	O
move	O
towards	O
the	O
root	O
of	O
the	O
treap	B-Application
and	O
speeding	O
up	O
future	O
accesses	O
for	O
the	O
same	O
keys	O
.	O
</s>
<s>
describe	O
a	O
sorting	B-Algorithm
algorithm	I-Algorithm
based	O
on	O
Cartesian	B-Algorithm
trees	I-Algorithm
.	O
</s>
<s>
They	O
describe	O
the	O
algorithm	O
as	O
based	O
on	O
a	O
tree	O
with	O
the	O
maximum	O
at	O
the	O
root	O
,	O
but	O
it	O
may	O
be	O
modified	O
straightforwardly	O
to	O
support	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
with	O
the	O
convention	O
that	O
the	O
minimum	O
value	O
is	O
at	O
the	O
root	O
.	O
</s>
<s>
The	O
Levcopoulos	O
–	O
Petersson	O
algorithm	O
can	O
be	O
viewed	O
as	O
a	O
version	O
of	O
selection	B-Algorithm
sort	I-Algorithm
or	O
heap	B-Application
sort	I-Application
that	O
maintains	O
a	O
priority	B-Application
queue	I-Application
of	O
candidate	O
minima	O
,	O
and	O
that	O
at	O
each	O
step	O
finds	O
and	O
removes	O
the	O
minimum	O
value	O
in	O
this	O
queue	O
,	O
moving	O
this	O
value	O
to	O
the	O
end	O
of	O
an	O
output	O
sequence	O
.	O
</s>
<s>
In	O
their	O
algorithm	O
,	O
the	O
priority	B-Application
queue	I-Application
consists	O
only	O
of	O
elements	O
whose	O
parent	O
in	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
has	O
already	O
been	O
found	O
and	O
removed	O
.	O
</s>
<s>
While	O
the	O
priority	B-Application
queue	I-Application
is	O
non-empty	O
:	O
</s>
<s>
As	O
Levcopoulos	O
and	O
Petersson	O
show	O
,	O
for	O
input	O
sequences	O
that	O
are	O
already	O
nearly	O
sorted	O
,	O
the	O
size	O
of	O
the	O
priority	B-Application
queue	I-Application
will	O
remain	O
small	O
,	O
allowing	O
this	O
method	O
to	O
take	O
advantage	O
of	O
the	O
nearly-sorted	O
input	O
and	O
run	O
more	O
quickly	O
.	O
</s>
<s>
They	O
also	O
prove	O
a	O
lower	O
bound	O
stating	O
that	O
,	O
for	O
any	O
and	O
(	O
non-constant	O
)	O
,	O
any	O
comparison-based	O
sorting	B-Algorithm
algorithm	I-Algorithm
must	O
use	O
comparisons	O
for	O
some	O
inputs	O
.	O
</s>
<s>
The	O
problem	O
of	O
Cartesian	B-Algorithm
tree	I-Algorithm
matching	O
has	O
been	O
defined	O
as	O
a	O
generalized	O
form	O
of	O
string	B-Algorithm
matching	I-Algorithm
in	O
which	O
one	O
seeks	O
a	O
substring	O
(	O
or	O
in	O
some	O
cases	O
,	O
a	O
subsequence	O
)	O
of	O
a	O
given	O
string	O
that	O
has	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
of	O
the	O
same	O
form	O
as	O
a	O
given	O
pattern	O
.	O
</s>
<s>
Fast	O
algorithms	O
for	O
variations	O
of	O
the	O
problem	O
with	O
a	O
single	O
pattern	O
or	O
multiple	O
patterns	O
have	O
been	O
developed	O
,	O
as	O
well	O
as	O
data	B-General_Concept
structures	I-General_Concept
analogous	O
to	O
the	O
suffix	B-Architecture
tree	I-Architecture
and	O
other	O
text	O
indexing	O
structures	O
.	O
</s>
<s>
Cartesian	B-Algorithm
trees	I-Algorithm
were	O
introduced	O
and	O
named	O
by	O
.	O
</s>
<s>
The	O
name	O
is	O
derived	O
from	O
the	O
Cartesian	O
coordinate	O
system	O
for	O
the	O
plane	O
:	O
in	O
one	O
version	O
of	O
this	O
structure	O
,	O
as	O
in	O
the	O
two-dimensional	O
range	B-Data_Structure
searching	I-Data_Structure
application	O
discussed	O
above	O
,	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
for	O
a	O
point	O
set	O
has	O
the	O
sorted	O
order	O
of	O
the	O
points	O
by	O
their	O
-coordinates	O
as	O
its	O
symmetric	O
traversal	O
order	O
,	O
and	O
it	O
has	O
the	O
heap	B-Application
property	I-Application
according	O
to	O
the	O
-coordinates	O
of	O
the	O
points	O
.	O
</s>
<s>
described	O
both	O
this	O
geometric	O
version	O
of	O
the	O
structure	O
,	O
and	O
the	O
definition	O
here	O
in	O
which	O
a	O
Cartesian	B-Algorithm
tree	I-Algorithm
is	O
defined	O
from	O
a	O
sequence	O
.	O
</s>
<s>
Using	O
sequences	O
instead	O
of	O
point	O
coordinates	O
provides	O
a	O
more	O
general	O
setting	O
that	O
allows	O
the	O
Cartesian	B-Algorithm
tree	I-Algorithm
to	O
be	O
applied	O
to	O
non-geometric	O
problems	O
as	O
well	O
.	O
</s>
